biggl( vol. 40, no. 5, pp. a2858--a2882lexing/entropy.pdf · 2018. 10. 17. · entropic fourier...

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SIAM J. SCI. COMPUT. c\bigcirc 2018 Society for Industrial and Applied MathematicsVol. 40, No. 5, pp. A2858--A2882

AN ENTROPIC FOURIER METHOD FOR THE BOLTZMANNEQUATION\ast

ZHENNING CAI\dagger , YUWEI FAN\ddagger , AND LEXING YING\S

Abstract. We propose an entropic Fourier method for the numerical discretization of theBoltzmann collision operator. The method, which is obtained by modifying a Fourier--Galerkinmethod to match the form of the discrete velocity method, can be viewed both as a discrete velocitymethod and as a Fourier method. As a discrete velocity method, it preserves the positivity ofthe solution and satisfies a discrete version of the H-theorem. As a Fourier method, it allows oneto readily apply the FFT-based fast algorithms. A second-order convergence rate is validated bynumerical experiments.

Key words. Fourier method, discrete velocity method, Boltzmann equation, modified Jacksonfilter, H-theorem, positivity

AMS subject classifications. 65M70, 65R20, 76P05

DOI. 10.1137/17M1127041

1. Introduction. Gas kinetic theory describes the statistical behavior of a largenumber of gas molecules in the joint spatial and velocity space. It has been widelyused to model gases outside the hydrodynamic regime, for example in the field ofrarefied gas dynamics. Let f(t, x, v) be the mass density distribution of the particles,depending on the time t \in \BbbR +, position x \in \BbbR d (d \geq 2), and microscopic velocityv \in \BbbR d. Based on the molecular chaos assumption, the Boltzmann equation

(1.1)\partial f

\partial t+ v \cdot \nabla xf = \scrQ [f, f ], f(0, x, v) = f0(x, v)

for the evolution of f(t, x, v) was derived in [6] and has served as the fundamentalequation in gas kinetic theory. When modeling the binary interaction between theparticles, the Boltzmann collision operator \scrQ [f, f ] takes the form

(1.2) \scrQ [f, f ](v) =

\int \BbbR d

\int \BbbS d - 1

\scrB (v - v\ast , \omega ) [f(v\prime )f(v\prime \ast ) - f(v)f(v\ast )] dv\ast d\omega

for monatomic gases, where

v\prime =v + v\ast

2+

| v - v\ast | 2

\omega , v\prime \ast =v + v\ast

2 - | v - v\ast |

2\omega

are the postcollisional velocities of two particles with precollisional velocities v and v\ast ,and \omega is the angular parameter of the collision. Here the variables t and x are omittedfor simplicity, and we shall continue to do so when focusing only on the collision term.The collision kernel \scrB is nonnegative and usually takes the form

(1.3) \scrB (v - v\ast , \omega ) = b(| v - v\ast | , cos \theta ), cos \theta = | (v - v\ast ) \cdot \omega | /| v - v\ast | .

\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section April 24,2017; accepted for publication (in revised form) June 13, 2018; published electronically September11, 2018.

http://www.siam.org/journals/sisc/40-5/M112704.html\dagger Department of Mathematics, National University of Singapore, Singapore 119076 (matcz@

nus.edu.sg).\ddagger Department of Mathematics, Stanford University, Stanford, CA 94305 ([email protected]).\S Department of Mathematics and Institute for Computational and Mathematical Engineering,

Stanford University, Stanford, CA 94305 ([email protected]).

A2858

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http://www.siam.org/journals/sisc/40-5/M112704.html

mailto:[email protected]





ENTROPIC FOURIER METHOD FOR BTE A2859

The Boltzmann equation (1.1) guarantees that f(t, x, v) remains nonnegative ifthe initial value f(t = 0, x, v) is also nonnegative [28]. The symmetry of the collisionterm (1.2) and the fact dv dv\ast = dv\prime dv\prime \ast imply that for any function \psi (\cdot )(1.4)\int

\BbbR d

\psi (v)\scrQ [f, f ](v) dv

=1

4

\int \BbbR d

\int \BbbR d

\int \BbbS d - 1

(\psi (v) + \psi (v\ast ) - \psi (v\prime ) - \psi (v\prime \ast ))\scrB [f(v\prime )f(v\prime \ast ) - f(v)f(v\ast )] dv\ast dv d\omega .

Setting \psi (v) = 1, v, | v| 2 gives rise to the conservation of the mass, momentum, andenergy

(1.5)

\int \BbbR d

\scrQ [f, f ](v) dv = 0,

\int \BbbR d

\scrQ [f, f ](v)v dv = 0,

\int \BbbR d

\scrQ [f, f ](v)| v| 2 dv = 0,

respectively. The famous H-theorem that states the monotonicity of the entropy

(1.6)

\int \BbbR d

\scrQ [f, f ](v) ln(f(v)) dv

=1

4

\int \BbbR d

\int \BbbR d

\int \BbbS d - 1

ln

\biggl( f(v)f(v\ast )

f(v\prime )f(v\prime \ast )

\biggr) \scrB [f(v\prime )f(v\prime \ast ) - f(v)f(v\ast )] dv\ast dv d\omega \leq 0

can also be obtained by setting \psi (v) = ln(f(v)).By introducing new variables y = v\prime - v and z = v\prime \ast - v and carrying out algebra

calculations, the Boltzmann collision operator can be rewritten as (see [7, 22, 19] fordetails)

(1.7) \scrQ [f, f ](v) =

\int \BbbR d

\int \BbbR d

\~\scrB (y, z)\delta (y \cdot z)[f(v + y)f(v + z) - f(v)f(v + y + z)] dy dz.

This is the well-known Carleman representation [7] of the Boltzmann collision opera-tor, where \~\scrB (y, z) is related to \scrB (v - v\ast , \omega ) in (1.3) by

(1.8) \~\scrB (y, z) = 2d - 1\scrB \biggl( y + z,

y - z

| y - z|

\biggr) | y + z| 2 - d.

Though the Boltzmann equation serves as the fundamental equation in gas dy-namics, its high-dimensional nature and the complexity of the collision operator posedifficulties for its numerical solution. A classical method is the direct Monte Carlosimulation [2], which uses simulated particles to mimic gas molecules and handles thecollisions in a stochastic way. Though it treats the high dimensionality effectively, theconvergence order is low and the numerical solution appears rather oscillatory.

With the rapid growth of computing power, it has become more practical to solvethe Boltzmann equation with deterministic methods. For all deterministic approaches,the complexity of the collision integral poses the most serious difficulty for numericalcomputation. Therefore, this paper focuses on the spatially homogeneous case

(1.9)\partial f

\partial t= \scrQ [f, f ]

for simplicity.In the past decades, several deterministic schemes have been developed for the

Boltzmann collision term. Two methods that have attracted the most attention arethe discrete velocity method (DVM) [13, 27, 4, 22] and the Fourier--Galerkin method(FGM) [5, 23, 24], which will be reviewed in what follows.

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A2860 ZHENNING CAI, YUWEI FAN, AND LEXING YING

1.1. Discrete velocity method. The DVM assumes that the particle velocitytakes only values from a finite set. Consider a domain \scrD T = [ - T, T ]d for the velocityvariable v that is discretized uniformly with step size h = 2T/N for a positive integerN (which is assumed to be odd for simplicity). By adopting the d-dimensional multi-index notation k = (k1, . . . , kd), one can denote the set of discrete velocity samplesby

(1.10) X = \{ h \cdot k| k = (k1, . . . , kd), - n \leq k1, . . . , kd \leq n\} ,

where N = 2n+ 1. In the rest of this paper, we use the lowercase letters p, q, r, ands to denote the discrete velocity samples in X.

Using Fr(t) for r \in X as the numerical approximations of the distribution functionf(t, v) at the points in X, the governing equations of DVM for Fr(t) are

(1.11)dFr(t)

dt= Qr(t) :=

\sum p,q,s\in X

Arspq (Fp(t)Fq(t) - Fr(t)Fs(t)) , r \in X.

Here Qr(t) serves as an approximation to \scrQ [f, f ](t, r). The coefficients Arspq are non-

negative constants and satisfy the conservation relations

(1.12) Arspq \not = 0 \Rightarrow p+ q = r + s and | p| 2 + | q| 2 = | r| 2 + | s| 2

and the symmetry property

(1.13) Arspq = Ars

qp = Apqrs .

The property (1.12) shows that the collisions in DVM also satisfy the momen-tum and energy conservation, and the property (1.13) implies that the collisions arealso reversible as in the Boltzmann equation. These two facts guarantee that DVMmaintains a number of fundamental physical properties of the continuous Boltzmannequation, such as (a) the positivity of the distribution function, (b) the exact conser-vation of mass, momentum, and energy, and (c) a discrete H-theorem.

More precisely, the values Fr(t) for r \in X are always nonnegative if the initialvalues Fr(t = 0) are nonnegative [28, 26]. The symmetry relation (1.13) implies that

(1.14)\sum r\in X

Qr\psi r =1

4

\sum p,q,r,s\in X

Arspq (\psi r + \psi s - \psi p - \psi q) (FpFq - FrFs) .

Combining (1.14) and the relations (1.12) gives rise to the conservation of mass,momentum, and energy in the discrete sense:

(1.15)\sum r\in X

Qr = 0,\sum r\in X

Qrr = 0,\sum r\in X

Qr| r| 2 = 0.

After letting \psi r = ln(Fr) in (1.14), one obtains a discrete version of the H-theoremfor DVM,

(1.16)\sum r\in X

Qr ln(Fr) =1

4

\sum p,q,r,s\in X

Arspq ln

\biggl( FrFs

FpFq

\biggr) (FpFq - FrFs) \leq 0,

using the nonnegativity of the coefficients Arspq and the monotonicity of the ln function.

Notice that, in this argument, the symmetry relations (1.13), the nonnegativity of

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Arspq, and the nonnegativity of the initial values are all essential for derivation of the

H-theorem.As a direct discretization for the high-dimensional integral of the collision term,

DVM has a rather high computational cost O(N2D+\delta ) for some 0 < \delta \leq 1 [8]. Itis also difficult to achieve a decent convergence rate due to the insufficient collisionpairs in the Cartesian grid used for velocity discretization. More precisely, in the 2Dcase, the rate of convergence of DVM introduced in [13] is only O((1/ log h)p) withp < 1/2 - 1/\pi [9]. For the 3D case, the best rate of convergence of DVM is also slowerthan the first order [21, 22].

The method in [18, 29] tries to improve the accuracy of DVM by interpolation.While the mass, momentum, and energy are conserved in this scheme, positivity andthe H-theorem fail to hold. The fast algorithm in [20] reduces the computationalcost of DVM to O( \=NdNd log(N)) with some parameter \=N \ll N for the hard spheremolecules, but it abandons the conservation of momentum and energy.

1.2. Fourier--Galerkin method. The Fourier-based methods assume that thedistribution function f(t, v) is supported (in the v variable) in a ball BR/2 centeredat the origin with radius R/2. Under this assumption, it makes sense to focus on thefunctions f(t, v) with Supp(f) \subset BR/2. For those functions, Supp(\scrQ [f, f ]) \subset B\surd

2R/2

and the collision term \scrQ [f, f ] reduces to a truncated version \scrQ R[f, f ] defined as

(1.17) \scrQ R[f, f ](v) :=

\int BR

\int BR

\~\scrB (y, z)\delta (y \cdot z)[f(v+y)f(v+z) - f(v)f(v+y+z)] dy dz,

where the superscript in \scrQ R[f, f ] denotes the truncation radius. In order to obtaina spectral approximation to the collision term, one restricts the domain of the distri-

bution function f(v) to the cube \scrD T = [ - T, T ]d with T \geq 3\surd 2+14 R in order to reduce

aliasing. One then extends it periodically to the whole space. (See [24, 19] for detailsof the derivation.) After periodization, f(v) can be written as a Fourier series

(1.18) f(v) =\sum k\in \BbbZ d

\^fkEk(v), \^fk =1

(2T )d

\int \scrD T

f(v)E - k(v) dv,

where Ek(v) = exp\bigl( \bfi \pi T k \cdot v

\bigr) . Substituting (1.18) into (1.17) gives rise to the following

representation of the truncated collision operator:

(1.19) \scrQ R[f, f ](v) =\sum

l,m\in \BbbZ d

\Bigl( \^B(l,m) - \^B(m,m)

\Bigr) \^fl \^fmEl+m(v), v \in \scrD T ,

where

(1.20) \^B(l,m) :=

\int BR

\int BR

\~\scrB (y, z)\delta (y \cdot z)El(y)Em(z) dy dz, l,m \in \BbbZ d.

It is easy to check that the coefficients \^B(l,m) are real and satisfy the symmetryrelations

(1.21) \^B(l,m) = \^B(m, l) = \^B(l, - m).

In terms of the Fourier expansion,

(1.22) \^\scrQ R[f, f ]k =\sum

l,m\in \BbbZ d

1(l +m - k)\Bigl( \^B(l,m) - \^B(m,m)

\Bigr) \^fl \^fm, k \in \BbbZ d.

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Here 1(\cdot ) is the indicator function, equal to 1 at the origin and 0 otherwise.The FGM in the literature (see, e.g., [24]) starts with a finite square grid of Fourier

modes

(1.23) K := \{ k| k = (k1, . . . , kd), - n \leq k1, . . . , kd \leq n\}

and the subspace

(1.24) \BbbP N = span \{ Ek(v)| k \in K\} \subset L2per(\scrD T ).

We shall denote the grid points in K with lowercase letters j, k, l, m. FGM approxi-mates the collision term (1.22) by projecting it to the subspace \BbbP N ,

(1.25) \^Q\sansG k :=

\sum l,m\in K

1(l +m - k)\Bigl( \^B(l,m) - \^B(m,m)

\Bigr) \^Fl\^Fm, k \in K,

where \^Fk for k \in K serve as the approximation of the Fourier modes \^fk of the exactsolution.

Putting together the above discussion, one arrives at the equations of the discreteFourier coefficients \^Fk(t) for k \in K for FGM [24],

(1.26)

\left\{ d \^Fk

dt= \^Q\sansG

k =\sum

l,m\in K

1(l +m - k)\Bigl( \^B(l,m) - \^B(m,m)

\Bigr) \^Fl\^Fm,

\^Fk(t = 0) = \^F 0k ,

where \^F 0k are the Fourier coefficients of the initial condition f0(v) restricted on \scrD T .

Remark 1.1. The above description of the FGM is based on the Carleman repre-sentation of the Boltzmann collision operator (1.7). Starting from the classical form(1.2), one can also derive a relation similar to (1.25) (see [24] for details), while thedefinition of \^B(l,m) is slightly different:

(1.27) \^B(l,m) =

\int BR

\int \BbbS d - 1

\scrB (g, \omega )El

\biggl( 1

2(g + | g| \omega )

\biggr) Em

\biggl( 1

2(g - | g| \omega )

\biggr) dg d\omega ,

where T \geq 3+\surd 2

4 R in the definition of Ek(\cdot ). It is straightforward to check that thesecoefficients also satisfy the symmetry relation (1.21).

FGM achieves spectral accuracy, although the computational cost is still as highas O(N2d) [24]. Two fast algorithms [19, 11] reduced the cost to O(MNd log(N))for the hard sphere molecules (the Maxwell molecules for the 2D case) [19] and toO(MNd+1 log(N)) for general collision kernels [11], where M is the number of pointsused for discretizing the unit sphere \BbbS d - 1.

Compared to DVM, the solution of FGM loses most of the aforementioned physicalproperties, including positivity, the conservation of momentum and energy, and the H-theorem. In [25], Pareschi and Russo proposed a positivity-preserving regularizationof FGM by using the Fej\'er filter at the expense of spectral accuracy. Despite this, thesolution fails to satisfy the H-theorem. The loss of the conservation can be fixed by aspectral Lagrangian strategy [12].

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1.3. Motivation. DVM preserves a number of physical properties (such as posi-tivity of the solution, the H-theorem, and exact conservation of mass, momentum, andenergy) but suffers from high computational costs and low accuracies. FGM enjoysspectral accuracies and lower computational costs but sacrifices almost all physicalproperties except the mass conservation. In this paper, we aim for a tradeoff betweenthe physical properties and the spectral accuracy.

As a fundamental property of the solution to the Boltzmann equation, the posi-tivity of the distribution function helps establish the H-theorem, which is one of theproperties crucial to guaranteeing the well-posedness of the discrete system. There-fore, it makes sense to maintain the positivity and the H-theorem, as long as it doesnot significantly sacrifice other properties such as numerical accuracy and efficiency.This paper is an initial study in this direction.

To achieve this goal, we first carefully study the reason behind the loss of theH-theorem in FGM by comparing it with DVM. With a few novel modifications toFGM, we propose an entropic Fourier method (EFM) that preserves the positivity,the mass conservation, and the H-theorem. In addition, the computational cost ofthis new method is the same as that of FGM.

The rest of the paper is organized as follows. In section 2, we first outline the keysteps in developing EFM and state the main results of the paper. The details of thederivation and some deeper understandings of the model are provided in section 3.Section 4 presents the implementation of EFM and the numerical results. The paperends with a discussion in section 5.

2. Main result. This section outlines the overall procedure of our derivationand lists some key results. Detailed derivation and investigation will be given insection 3.

Aiming at developing an entropic Fourier method for the homogeneous Boltzmannequation, one works mainly with the evolution of the Fourier coefficients \^Fk(t). Recallthe discrete Fourier transform

(2.1) Fp =\sum k\in K

\^FkEk(p), \^Fk =1

Nd

\sum p\in X

FpE - k(p),

where X defined in (1.10) and K defined in (1.23) are the sets of uniform samples inthe velocity space and the Fourier domain, respectively.

Using the discrete Fourier transform, one can instead treat the point values Fp

as the degrees of freedom and write the numerical scheme in the DVM form (1.11).According to the derivation in section 1.1, the following condition is required in orderto guarantee the H-theorem for DVM.

Condition 1. The DVM defined in (1.11) satisfies the following:1. the coefficients Ars

pq satisfy the symmetry relation Arspq = Ars

qp = Apqrs;

2. the coefficients Arspq are nonnegative, i.e., Ars

pq \geq 0;3. the initial values are nonnegative, i.e., Fp(t = 0) \geq 0 for any p \in X.

The general idea of our approach is to revise the existing FGM so that Condition1 is fulfilled. Below we list the steps that lead to a numerical scheme that satisfiesthe H-theorem.

1. Apply the Fourier collocation method to (1.19). This leads to an approxima-tion to (1.22) in the form

(2.2) \^Q\sansC k =

\sum l,m\in K

1N (l +m - k)[ \^BN (l,m) - \^BN (m,m)] \^Fl\^Fm, k \in K,

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where 1N (l) := 1(l mod N) and \^BN (l,m) := \^B(l mod N,m mod N). Heremod is the symmetric modulo function, i.e., each component of l mod Nranging from - n to n (recall N = 2n + 1). Using the relation between theFourier coefficients and the values on collocation points (2.1), we can rewrite(2.2) as

(2.3) Q\sansC r =

\sum p,q,s\in X

Arspq[FpFq - FrFs], r \in X,

where Arspq (given in (3.8)) is determined by the Fourier modes of the collision

kernel \^BN (\cdot , \cdot ) and satisfies the symmetry relation (Condition 1.1).2. A careful study shows that Ars

pq fails to be nonnegative. This can be fixed by

applying a positivity-preserving filter to \^BN (l,m), i.e.,

(2.4) \^B\sigma N (l,m) := \^BN (l,m)\sigma N (l)\sigma N (m), l,m \in K,

where \sigma N (l) is the tensor-product of d, the one-dimensional modified Jacksonfilter [17, 30]. The modified collision term takes the following form in theFourier domain:

\^Q\sigma k =

\sum l,m\in K

1N (l +m - k)[ \^B\sigma N (l,m) - \^B\sigma

N (m,m)] \^Fl\^Fm, k \in K.

Using (A\sigma )rspq to denote the coefficients determined by the new kernel modes

\^B\sigma N (l,m) and writing

Q\sigma r =

\sum p,q,s\in X

(A\sigma )rspq[FpFq - FrFs], r \in X,

one can verify that both the symmetry relation (Condition 1.1) and the non-negativity (Condition 1.2) are satisfied.

3. To guarantee the positivity of the initial values (Condition 1.3), we adoptinterpolation rather than orthogonal projection while discretizing the initialdistribution function.

Main result. Summarizing the outline given above, we arrive at a new entropicFourier method (EFM) that takes the following simple form:

(2.5)

\left\{ d \^Fk

dt= \^Q\sigma

k =\sum

l,m\in K

1N (l +m - k)\Bigl( \^B\sigma N (l,m) - \^B\sigma

N (m,m)\Bigr) \^Fl\^Fm,

\^Fk(t = 0) =1

Nd

\sum r\in X

f(t = 0, r)E - k(r).

This method preserves several key physical properties, as guaranteed by the followingtheorem.

Theorem 2.1. If f(t = 0, v) \geq 0 for v \in \BbbR d, then the solution Fr(t) =\sum

k\in K\^Fk(t)

\cdot Ek(r) for r \in X of (2.5) satisfies for all t > 0

conservation of mass:d

dt

\sum r\in X

Fr(t) = 0,(2.6)

nonnegativity: Fr(t) \geq 0, r \in X,(2.7)

discrete H-theorem:d

dt

\sum r\in X

Fr(t) lnFr(t) \leq 0.(2.8)

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The proof is presented in section 3.5. Due to the positivity-preserving filter (3.21),the numerical accuracy of EFM in approximating the collision operator is second-order(see section 3.4 for details).

Another important result for EFM is the existence of fast algorithms. For FGM,the fast algorithms proposed in [19, 11] are based on the approximation of the kernel\^BN (\cdot , \cdot ):

(2.9) \^BN (l,m) \approx M\sum t=1

\alpha tl+m\beta

tl\gamma

tm, l,m \in K,

with the number of terms M \ll Nd. Since the filtered kernel \^B\sigma N (l,m) turns out to

have a similar approximation,

(2.10) \^B\sigma N (l,m) \approx

M\sum t=1

\alpha tl+m

\bigl( \sigma N (l)\beta t

l

\bigr) \bigl( \sigma N (m)\gamma tm

\bigr) ,

these fast algorithms still apply. Moreover, when the above approximation is applied,the H-theorem still holds. Detailed discussion will be given in section 4.1.

3. Entropic Fourier method. As shown in section 1.1, a discrete H-theoremcan be obtained from the classical DVM, where the associated entropy function canbe considered as a numerical quadrature for the integral of f ln f . This requires thepositivity of the distribution function, which can be guaranteed by the positivity ofthe discrete collision kernel Ars

pq. In general, to preserve the Boltzmann entropy in thenumerical scheme, the positivity of the numerical solution needs to be enforced in acertain sense due to the presence of ln f in the entropy function. However, in FGM,there is no guarantee of any form of positivity in the numerical solution, and hencethe H-theorem does not hold.

In this paper, rather than enforcing the nonnegativity of the whole distributionfunction, we take a collocation approach and focus on the nonnegativity only atthe collocation points. Based on this idea, we start from a collocation method forthe homogeneous Boltzmann equation and write it as a DVM of the function valuesdefined at the collocation points. One then tries to alter the coefficients to match therequirements in Condition 1 so that the H-theorem can be subsequently derived.

The three steps listed in section 2 are detailed in the first three subsections be-low. After that, section 3.4 compares the entropic Fourier method (EFM) with otherFourier methods.

3.1. Fourier collocation method in a DVM form. The mechanisms of DVMand FGM are quite different: DVM is concerned with the values of the distributionon discrete points, whereas FGM (1.25), as a Galerkin method, works on the Fouriermodes of the distribution function. It is not straightforward how to link these twomethods. Alternatively, we will consider another type of Fourier methods---the collo-cation method (also known as the pseudospectral method).

3.1.1. Fourier collocation method. In the Fourier collocation method (FCM),the collision term on the set X is evaluated directly using (1.19):

(3.1) Q\sansC r =

\sum l,m\in K

\Bigl( \^BN (l,m) - \^BN (m,m)

\Bigr) \^Fl\^FmEl+m(r), r \in X,

where \^BN (l,m) := \^B(l mod N,m mod N), and mod is the symmetric modulo func-tion, i.e., each component of l mod N ranging from - n to n (recall N = 2n + 1).

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Since the above equation only uses the value of \^BN in K, one can use \^B and \^BN in-terchangeably here. Here we note that \^BN (l,m) satisfy the symmetry relation (1.21).

The corresponding Fourier modes can be obtained by an inverse discrete Fouriertransform:

(3.2)

\^Q\sansC k =

1

Nd

\sum r\in X

QrE - k(r)

=\sum

l,m\in K

1N (l +m - k)\Bigl( \^BN (l,m) - \^BN (m,m)

\Bigr) \^Fl\^Fm,

where 1N (l) := 1(l mod N).If the initial value is smooth enough, due to the smoothing effect of the Boltzmann

collision operator [1], both FGM and FCM have spectral accuracy [8]. Moreover, insome cases, FCM (3.2) is numerically more efficient, especially for the fast summationalgorithms in [19, 11]. For example, in [19], the following approximation of \^BN (l,m)is considered:

(3.3) \^BN (l,m) \approx M\sum t=1

\beta tl\gamma

tm,

where M \in \BbbN + is the total number of quadrature points on the sphere. Then thecollision term in this Galerkin method can be approximated by

(3.4)

M\sum t=1

\sum l,m\in K

1(l +m - k)\Bigl[ \Bigl( \beta tl\^Fl

\Bigr) \Bigl( \gamma tm \^Fm

\Bigr) - \^Fl

\Bigl( \beta tm\gamma

tm\^Fm

\Bigr) \Bigr] .

To evaluate (3.4) efficiently, one needs to utilize FFT-based convolutions. To obtainthese coefficients, one needs the zero-padding technique to avoid aliasing. If one usesthe same method to evaluate \^Qk in (3.2), then no zero-padding is needed. Therefore,the collocation method shortens the length of vectors used in the Fourier transform,which makes the algorithm faster.

3.1.2. DVM form. To link FCM with DVM, we split the collision term (3.1)into the gain part (+) and the loss part ( - ):(3.5)

Q\sansC ,+r =

\sum l,m\in K

\^BN (l,m) \^Fl\^FmEl+m(r), Q\sansC , -

r =\sum

l,m\in K

\^BN (m,m) \^Fl\^FmEl+m(r), r \in X.

Noticing El+m(r) =\sum

k\in K 1N (l+m - k)Ek(r) and plugging (2.1) into the gain partyields

(3.6) Q\sansC ,+r =

1

N2d

\sum l,m,k\in Kp,q\in X

1N (l +m - k) \^BN (l,m)E - l(p)E - m(q)Ek(r)FpFq.

Since 1Nd

\sum s\in X Ej(s) = 1N (j), one can sum over j to get 1

Nd

\sum j\in K,s\in X Ej(s) = 1.

With this equation, one introduces two new indices to (3.6) by multiplying its right-hand side with 1

Nd

\sum j\in K,s\in X Ej(s):

(3.7)

Q\sansC ,+r =

1

N3d

\sum l,m,k,j\in Kp,q,s\in X

1N (l+m - k - j) \^BN (l - j,m - j)E - l(p)E - m(q)Ek(r)Ej(s)FpFq.

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If one introduces

(3.8) Arspq =

1

N3d

\sum l,m,k,j\in K

1N (l+m - k - j) \^BN (l - j,m - j)E - l(p)E - m(q)Ek(r)Ej(s),

then the gain term is

(3.9) Q\sansC ,+r =

\sum p,q,s\in X

ArspqFpFq.

Apparently, such a term does take the form of the gain term of DVM (1.11).For the loss term, the identity

(3.10)\sum

p,q\in X

Arspq =

1

Nd

\sum k,j

1N (k + j) \^BN (j, j)Ek(r)Ej(s)

leads to the following derivation:(3.11)\sum p,q,s\in X

ArspqFrFs =

1

Nd

\sum k,j\in K,s\in X

1N (k + j) \^BN (j, j)Ek(r)Ej(s)FrFs

=1

Nd

\sum k,j,l,m\in K,s\in X

1N (k + j) \^BN (j, j)Ek(r)Ej(s)El(r)Em(s) \^Fl\^Fm

=\sum

k,j,l,m\in K

1N (k + j)1N (j +m) \^BN (j, j) \^Fl\^FmEk+l(r)

=\sum

l,m\in K

\^BN (m,m) \^Fl\^FmEl+m(r) = Q\sansC , -

r .

In summary, one can write FCM in the following DVM form:

(3.12) Q\sansC r =

\sum p,q,s\in X

Arspq(FpFq - FrFs)

with Arspq given in (3.8). Finally, the symmetry relation (Condition 1.1)

(3.13) Arspq = Ars

qp = Apqrs

holds, as this can be easily seen by the symmetry relation of \^BN (l,m) and switchingthe indices in (3.8).

3.2. Positivity preservation. As remarked earlier, in order to obtain an H-theorem for FCM, one needs to ensure that all the coefficients Ars

pq are nonnegative(Condition 1.2). Below, we first show that Ars

pq as defined in (3.12) fail to be non-negative, and then we apply a filter to recover nonnegativity.

3.2.1. Failure of positivity preservation in FCM. We start by simplifyingthe coefficients Ars

pq based on (3.8):(3.14)

Arspq =

1

N3d

\sum l,m,k,j\in K

1N (l +m - k - j) \^BN (l - j,m - j)E - l(p)E - m(q)Ek(r)Ej(s)

=1

N3d

\sum l,m,k\in K

\^BN (m - k, l - k)E - l(p - s)E - m(q - s)Ek(r - s),

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where one uses j = l + m - k mod N . By performing a change of variables i =(m - k) mod N and j = (l - k) mod N , we arrive at

(3.15)

Arspq =

1

N3d

\sum i,j,k\in K

\^BN (i, j)E - k - i(p - s)E - k - j(q - s)Ek(r - s)

= 1N (r + s - p - q)1

N2d

\sum i,j\in K

\^BN (i, j)E - i(p - s)E - j(q - s).

By introducing

(3.16) G(y, z) =\sum

i,j\in K

\^BN (i, j)E - i(y)E - j(z), y, z \in \scrD T ,

which is by definition a periodic function with period \scrD T , one can write compactly

(3.17) Arspq =

1

N2d1N (r + s - p - q)G(p - s, q - s), p, q, r, s \in X.

In order to check whether Arspq is nonnegative, one just needs to check whether

G(\cdot , \cdot ) is nonnegative on the collocation points in X. To get a better understanding ofthe function G(\cdot , \cdot ) as defined in (3.16), one applies the definition of \^BN (\cdot , \cdot ) to obtain

(3.18) G(y, z) =

\int BR

\int BR

\~\scrB (y\prime , z\prime )\delta (y\prime \cdot z\prime )\chi N (y - y\prime )\chi N (z - z\prime ) dy\prime dz\prime .

Here \chi N is the Dirichlet kernel over \scrD T defined by

(3.19) \chi N (v) =\sum k\in K

Ek(v), v \in \scrD T ,

and its discrete Fourier transform \^\chi N (k) is equal to 1 for k \in K and 0 on \BbbZ d \setminus K. Byintroducing a periodic function in \scrD T

(3.20) H(y, z) = \~\scrB (y, z)\delta (y \cdot z)1(| y| \leq R)1(| z| \leq R), y, z \in \scrD T ,

one can write

G = H \ast (\chi N \otimes \chi N ),

where the convolution is defined periodically in \scrD T \times \scrD T . Equivalently, G(y, z) is alsothe truncated Fourier expansion of H(y, z) by keeping only the frequencies in K.

Although H(y, z) is nonnegative in the weak sense, its truncated Fourier approxi-mation G(y, z) fails to be so. For example, the values of G for the kernel \~\scrB (y, z) \equiv 1

\pi ,R = 6 in 2D are plotted in Figure 1. This clearly shows that negative values appearas expected. Therefore, in general, the H-theorem does not hold for FCM.

3.2.2. Filtering. In the previous subsection, one can see that if \chi N (\cdot ) were anonnegative function, then G(y, z) would be nonnegative for any y and z. Thus, inorder to get nonnegative coefficients, a possible way is to replace the function \chi N bya nonnegative one. Note that \chi N (v) is a Dirichlet kernel, which is an approximationof the Dirac delta. As N \rightarrow +\infty , the function \chi N (\cdot ) tends to the Dirac delta weaklyin an oscillatory way. As pointed out in [10], the oscillation breaks the nonnegativityof the solution.

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-2

0

5

9

13

Fig. 1. The values of G(y, z) with N = 32 at z = (T/2, T/2). The axes are the two componentsof y.

As mentioned earlier in section 2, we adopt the one-dimensional modified Jacksonfilter [17, 30] given by

(3.21) \sigma N (\beta ) =(n+ 1 - | \beta | ) cos( \pi | \beta | n+1 ) + sin( \pi | \beta | n+1 ) cot(

\pi n+1 )

n+ 1,

where N = 2n+1 and - n \leq \beta \leq n. By a slight abuse of notation, the d-dimensionalmodified Jackson filter for a multi-index k = (k1, . . . , kd) \in K is defined throughtensor product

\sigma N (k) =

d\prod i=1

\sigma N (ki).

The modified kernel \chi \sigma N (v) can then be defined as

\chi \sigma N (v) =

\sum k\in K

\sigma N (k)Ek(v), v \in \scrD T .

Once \chi N is replaced with \chi \sigma N in (3.18), the function G(y, z) is substituted with

G\sigma (y, z) := (G \ast (\chi \sigma N \otimes \chi \sigma

N ))(y, z). A direct calculation shows that(3.22)

G\sigma (y, z) =1

N2d

\int BR

\int BR

\~\scrB (y\prime , z\prime )\delta (y\prime \cdot z\prime )\chi \sigma N (y - y\prime )\chi \sigma

N (z - z\prime ) dy\prime dz\prime

=1

N2d

\sum l,m\in K

\int BR

\int BR

\~\scrB (y\prime , z\prime )\delta (y\prime \cdot z\prime )\sigma N (l)\sigma N (m)El(y - y\prime )Em(z - z\prime ) dy\prime dz\prime

=1

N2d

\sum l,m\in K

\Bigl[ \sigma N (l)\sigma N (m) \^BN (l,m)

\Bigr] E - l(y)E - m(z)

=1

N2d

\sum l,m\in K

\^B\sigma N (l,m)E - l(y)E - m(z),

where \^B\sigma N (l,m) := \^BN (l,m)\sigma N (l)\sigma N (m) as defined in (2.4). With G\sigma (y, z) \geq 0

guaranteed, one can mimic (3.17) and define

(3.23) (A\sigma )rspq =1

N2d1N (r + s - p - q)G\sigma (p - s, q - s), p, q, r, s \in X,

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which are apparently nonnegative. Since replacing \^BN (l,m) with \^B\sigma N (l,m) does not

affect the symmetry relation (1.21), the new coefficients (A\sigma )rspq also satisfy (A\sigma )rspq =(A\sigma )rsqp = (A\sigma )pqrs .

From the above discussion, we now define the entropic collision term to be

(3.24) \^Q\sigma k :=

\sum l,m\in K


N (m,m)\Bigr) \^Fl\^Fm, k \in K,

in the Fourier domain. In the velocity domain, it is equal to

(3.25) Q\sigma r :=

\sum p,q,s\in X

(A\sigma )rspq (FpFq - FrFs) , r \in X.

3.3. Initial condition. In order to obtain the H-theorem, one needs to makesure that the initial data are nonnegative. Since the collision term in (3.25) dependsonly on the values at the collocation points inX, it is sufficient to make the initial datanonnegative at these collocation points. Consequently, it is natural to use samplingrather than orthogonal projection while preparing the discrete initial data F 0

r forr \in X. More precisely,

(3.26) F 0r = \scrI Nf0 :=

\Biggl\{ f0(r), f0 is continuous,

(\varphi \epsilon \ast f0)(r) otherwise,

where \varphi \epsilon \geq 0 is a mollifier, such that \| f0 - \varphi \epsilon \ast f0\| L2 < \epsilon for \epsilon sufficiently small.Once \{ F 0

r \} are ready, the corresponding Fourier coefficients \{ \^F 0k \} are computed via a

fast Fourier transform.At this point, all ingredients of the entropic Fourier method (EFM) are ready.

The Cauchy problem of EFM takes the following form as a DVM:

(3.27)

\left\{ dFr

dt= Q\sigma

r =\sum

p,q,s\in X

(A\sigma )rspq (FpFq - FrFs) ,

Fr(t = 0) = F 0r .

Equivalently, in the Fourier domain, EFM takes the form

(3.28)

\left\{ d \^Fk

dt= \^Q\sigma

k =\sum

l,m\in K


N (m,m)\Bigr) \^Fl\^Fm,

\^Fk(t = 0) = \^F 0k .

Remark 3.1. The same technique can be applied to the Fourier method derivedfrom the classical form of the Boltzmann collision operator (1.2) to obtain an entropicFourier method. In fact, from (1.27) and following the definition of G(y, z) in (3.16),one can directly obtain

G(y, z) = (H\ast (\chi N\otimes \chi N ))(y, z), H(g, g\prime ) = \scrB (g, \omega )\delta (| g| - | g\prime | )1(| g| \leq R), g, g\prime \in \scrD T .

Again, by (3.17), the positivity of Arspq depends only on the positivity of G(y, z) at

the collocation points. Therefore, replacing \chi N with \chi \sigma N does the job.

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3.4. Comparison. The derivation of EFM switched frequently between the lan-guage of Fourier methods and DVM for different purposes. As we have shown in (3.27)and (3.28), EFM can be regarded either as a Fourier method or as a special DVM.

In what follows, we provide a comparison between the EFM (entropic Fouriermethod), the FGM (Fourier--Galerkin method), and the Fourier collocation method(FCM). To set up a uniform notation, let \scrQ [\cdot ; \cdot , \cdot ] be the general collision operator

(3.29) \scrQ [C; f, f ](v) =

\int \scrD T

\int \scrD T

C(y, z)[f(v + y)f(v + z) - f(v)f(v + y + z)] dy dz

with a collision kernel C(\cdot , \cdot ). Thus the truncated collision term (1.17) can be writtenas \scrQ (H; f, f) using the definition of H in (3.20).

Notice that a special feature of a function in \BbbP N is that it is uniquely defined viaits function values at points in X defined in (1.10). Therefore, \{ Fp| p \in X\} can beregarded as both a discrete set of values and the samples from the smooth periodicF (v) \in \BbbP N \subset L2

per(\scrD T ). By introducing two operators

\scrP N : f \rightarrow \chi N \ast f, \scrS \sigma N : f \rightarrow \chi \sigma

N \ast f

for the space L2per(\scrD T ), the three methods are different approximations of \scrQ (H; f, f)

with different initial values:

FGM: \scrP N\scrQ [(\scrP N \otimes \scrP N )H;F, F ], F (t = 0, v) = \scrP Nf(t = 0, v),(3.30)

FCM: \scrI N\scrQ [(\scrP N \otimes \scrP N )H;F, F ], F (t = 0, v) = \scrP Nf(t = 0, v),(3.31)

EFM: \scrI N\scrQ [(\scrS \sigma N \otimes \scrS \sigma

N )H;F, F ], F (t = 0, v) = \scrI Nf(t = 0, v),(3.32)

where \scrI N is the interpolation operator defined in (3.26).The list (3.30)--(3.32) clearly shows how we change from FGM to EFM in our

derivation. The last line (3.32) also shows that EFM provides an approximation ofthe original binary collision operator in the language of spectral methods. Below wewill briefly review the basic properties of all three methods.

The method (3.30) stands for FGM as described in section 1.2. In the derivation,the kernel K is not explicitly projected. However, (1.25) shows that the discretecollision operator depends only on (\scrP N \times \scrP N )H. By replacing the projection operatorapplied to \scrQ with interpolation as in (3.2), we arrive at the Fourier collocation method(3.31) introduced in section 3.1. Since a direct projection of H does not preserve thepositivity of the kernel, the negative part of the discrete kernel may cause a violationof the H-theorem. Nevertheless, both of these methods have spectral accuracy in thevelocity space.

To ensure the positivity of the discrete kernel, the filter \scrS \sigma N \otimes \scrS \sigma

N is applied in(3.32), and thus positive coefficients (3.23) are obtained. The method (3.32) alsoensures the positivity of the approximation of F at collocation points, and thus thediscrete H-theorem follows.

However, the filter S\sigma N has a smearing effect, which reduces the order of conver-

gence. For any smooth periodic function f \in L2per(\scrD T ), the L

2-error \| f - \scrS \sigma Nf\| 2 is

O(N - 2) [16, Chapter 4], and therefore EFM is at most second-order. On the otherhand, if one splits the collision term of EFM into the gain part and the loss part and

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lets F = \scrP Nf , then

Q\sigma ,+[F, F ](r) =\sum

l,m\in K

\^BN (l,m)\sigma N (l) \^Fl\sigma N (m) \^FmEl+m(r) = Q\sansC ,+[\scrS \sigma NF,\scrS \sigma

NF ](r),

(3.33)

Q\sigma , - [F, F ](r) =\sum

l,m\in K

\^BN (m,m) \^Fl\sigma 2N (m) \^FmEl+m(r) = Q\sansC , - [F,\scrS \sigma

N\scrS \sigma NF ](r)

(3.34)

for any r \in X. Following the boundedness of the truncated collision operator provenin [24], one concludes that

\| Q\sigma [F, F ] - Q\sansC [F, F ]\| 2 \leq \| Q\sigma ,+[F, F ] - Q\sansC ,+[F, F ]\| 2+\| Q\sigma , - [F, F ] - Q\sansC , - [F, F ]\| 2 \leq O(N - 2).

Hence, EFM has second-order accuracy in approximating the truncated collision op-erator if the distribution is smooth enough (due to the smoothing effect of the Boltz-mann collision operator [1], we only need that the initial value is smooth enough).This order of convergence will be also numerically verified in the next section.

3.5. Proof of Theorem 2.1. This subsection provides the proof of Theorem 2.1.

Proof of Theorem 2.1. The argument in section 3.2.2 indicates that if f(t =0, v) \geq 0, then the coefficients Ars

pq and the initial values F 0r satisfy all three con-

ditions in Condition 1.The symmetry relation (1.21) of \^B(l,m) and the definition of \^B\sigma

N (l,m) indicate

1N (l+m)( \^B\sigma N (l,m) - \^B\sigma

N (m,m)) = 0, i.e., Q\sigma 0 = 0. Noticing that the zero frequency

\^F0(t) =1

Nd

\sum r\in X Fr(t), one can directly obtain the conservation of mass (2.6).

Since f0(v) \geq 0, F 0r \geq 0 by construction. If there exist t\prime > 0 and r \in X such

that Fr(t\prime ) = 0 and Fp(t

\prime ) \geq 0 for any other p \in X, then

dFr(t)

dt| t=t\prime =

\sum p,q,s\in X

(A\sigma )rspqFpFq \geq 0,

which indicates Fr(t) \geq 0 for all t > 0 and r \in X.The symmetry relation and nonnegativity of (A\sigma )rspq indicate the discrete H-

theorem (2.8)\sum r\in X

Q\sigma r ln(Fr) =

1

4

\sum p,q,r,s\in X

(A\sigma )rspq ln

\biggl( FrFs

FpFq

\biggr) [FpFq - FrFs] \leq 0.

This completes the proof.

4. Numerical tests. This section describes several numerical tests to demon-strate the properties of EFM and to compare with the FGM in [24] and the positivity-preserving spectral method (PPSM) in [25].

4.1. Implementation. It is pointed out in section 2 that the fast algorithmsin [19, 11] can be applied to EFM without affecting the H-theorem. To show this,one needs to check that the fast algorithms do not violate the first two conditions inCondition 1.

These fast algorithms are based on an approximation of \^BN (l,m) (2.10) of thefollowing form:

(4.1) \^BN (l,m) \approx \^BN,\sansf \sansa \sanss \sanst (l,m) :=

M\sum t=1

\alpha tl+m\beta

tl\gamma

tm.

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After the filtering is applied, one obtains a similar approximation for \^B\sigma N (l,m):

(4.2) \^B\sigma N (l,m) \approx \^B\sigma

N,\sansf \sansa \sanss \sanst (l,m) :=

M\sum t=1

\alpha tl+m

\bigl( \sigma N (l)\beta t

l

\bigr) \bigl( \sigma N (m)\gamma tm

\bigr) .

It can be verified that both kernels satisfy the symmetry relation

\^BN,\sansf \sansa \sanss \sanst (l,m) = \^BN,\sansf \sansa \sanss \sanst (m, l) = \^BN,\sansf \sansa \sanss \sanst ( - l,m),(4.3)

\^B\sigma N,\sansf \sansa \sanss \sanst (l,m) = \^B\sigma

N,\sansf \sansa \sanss \sanst (m, l) = \^B\sigma N,\sansf \sansa \sanss \sanst ( - l,m),(4.4)

which indicates that Condition 1.1 is valid for the fast algorithms.To see that the fast algorithms do not affect the nonnegativity of G\sigma (y, z), we use

the fast algorithm in [19] with d = 2 and \~B = 1 as an example. The first step of thisalgorithm writes y and z in (1.20) in the polar coordinates y = \rho e and z = \rho \ast e\ast :

(4.5) \^BN (l,m) =1

4

\int \BbbS 1

\int \BbbS 1\delta (e \cdot e\ast )

\Biggl[ \int R

- R

El(\rho e) d\rho

\Biggr] \Biggl[ \int R

- R

Em(\rho \ast e\ast ) d\rho \ast

\Biggr] dede\ast .

Let \psi R(l, e) =\int R

- REl(\rho e) d\rho ; then

\^BN (l,m) =1

4

\int \BbbS 1

\int \BbbS 1\delta (e \cdot e\ast )\psi R(l, e)\psi R(m, e\ast ) dede\ast .

Integrating it with respect to e\ast yields

(4.6) \^BN (l,m) =

\int \pi

0

\psi R(l, e\theta )\psi R(m, e\theta +\pi /2) d\theta .

Substituting (4.5) into (3.22) gives rise to(4.7)

G\sigma (y, z) =1

4

\int \BbbS 1

\int \BbbS 1\delta (e \cdot e\ast )

\Biggl[ \int R

- R

\chi \sigma N (\rho e - y) d\rho

\Biggr] \Biggl[ \int R

- R

\chi \sigma N (\rho e\ast - z) d\rho \ast

\Biggr] de de\ast .

Let \phi \sigma R(y, e) =\int R

- R\chi \sigma N (\rho e - y) d\rho . Apparently, \phi \sigma R(y, e) \geq 0 due to \chi \sigma

N (y) \geq 0 for any

y \in \BbbR 2. Then integrating (4.7) with respect to e\ast yields

(4.8) G\sigma (y, z) =

\int \pi

0

\phi \sigma R(y, e\theta )\phi \sigma R(z, e\theta +\pi /2) d\theta .

The idea of the fast algorithm is to replace the integration in (4.6) with a quadra-ture formula. More precisely, (4.6) is approximated by

(4.9) \^BN,\sansf \sansa \sanss \sanst (l,m) =

M\sum t=1

\pi

M\psi R(l, e\theta t)\psi R(m, e\theta t+\pi /2).

Similarly to (4.9), one obtains

(4.10) G\sigma \sansf \sansa \sanss \sanst (y, z) =

M\sum t=1

\pi

M\phi \sigma R(y, e\theta t)\phi

\sigma R(z, e\theta t+\pi /2).

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Since \phi \sigma R(y, e) \geq 0 for any y \in \BbbR 2, e \in \BbbS 1, G\sigma \sansf \sansa \sanss \sanst (y, z) \geq 0 for any y, z \in \BbbR 2. Hence,

the fast algorithm does not destroy the nonnegativity of G\sigma \sansf \sansa \sanss \sanst (y, z).

As we pointed out in section 3.1, an aliased convolution can be directly used tocalculate (2.5). Since the accuracy of EFM is only second-order, the smoothing filteris the main source of the error. In the fast algorithms, the numberM in (4.1) perhapscan be smaller than that in [19, 11].

In the above discussion, we only study the case when N is odd. The case of evenN values can be reduced to the odd (N - 1) case by setting the coefficient of a modek to be 0 if any component of k = (k1, . . . , kd) is equal to - N/2.

For the time discretization, the third-order strong stability-preserving Runge--Kutta method proposed in [14] is employed in the discretization of time. In all tests,the time step is chosen as \Delta t = 0.01.

4.2. Numerical results. The test problems used here are solutions of the space-homogeneous Boltzmann equation for Maxwell molecules (\scrB (g, \omega ) = 1

2\pi in 2D and\scrB (g, \omega ) = 1

4\pi in 3D).

Example 1 (2D BKW solution). The first example is the well-known 2D Bobylev--Krook--Wu (BKW) solution, obtained independently in [3] and [15]. The exact solu-tion takes the form

(4.11) f(t, v) =1

2\pi Sexp

\biggl( - | v| 2

2S

\biggr) \biggl( 2S - 1

S+

1 - S

2S2| v| 2

\biggr) ,

where S = 1 - exp( - t/8)/2. The BKW solution allows one to check the accuracy,positivity of the solution, and the entropy of the proposed method. Here we set thetruncation radius R = 6 in the tests.

0 2 4 6 8 10

-4

-3.5

-3

-2.5

-2

-1.5

-1

FGM P

FGM I

FCM P

FCM I

t

log10(\epsilon )

(a) N = 16

0 2 4 6 8 10

-14

-12

-10

-8

-6

-4

FGM P

FGM I

FCM P

FCM I

t

log10(\epsilon )

(b) N = 32

Fig. 2. Positivity error of FGM and FCM with the initial value prepared by orthogonal projec-tion (P) and interpolation (I) in the log10 scale. Since the positivity error of EFM is strictly zero,its result is not plotted in the figure.

Both FGM (1.26) and FCM (3.2) result in good approximations of the exactsolution at the collocation points. However, the solutions are not nonnegative. Evenif we use interpolation rather than orthogonal projection to prepare the initial values,the solutions of these methods still fail to be nonnegative. The positivity of the

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solution is measured by the positivity error defined via

(4.12) \epsilon :=

\sum q\in X | Fq| -

\sum q\in X Fq\sum

q\in X | Fq| .

Figure 2 shows that both FGM and FCM fail to preserve the positivity of the solutionat the collocation points regardless of whether the initial value is given by orthogonalprojection or interpolation. On the other hand, the positivity error of EFM is strictlyzero, thanks to the modification in (3.22).

Table 1 summarizes the \ell 1, \ell 2, and \ell \infty errors of (3.27) at time t = 0.01. Here the\ell p relative errors for p = 1, 2,\infty are defined by

(4.13)\| F - f\| p

\| f\| p=

\Bigl( \sum q\in X | Fq - f(q)| p

\Bigr) 1/p

\Bigl( \sum q\in X | f(q)| p

\Bigr) 1/p,

where f(q) is the exact solution at q \in X. The numerical results also show that theconvergence rate of EFM is of second order.

Table 1The \ell 1, \ell 2, and \ell \infty errors and convergence rates for the BKW solution at time t = 0.01 with

R = 6.

N \ell 1 error Rate \ell 2 error Rate \ell \infty error Rate16 4.68\times 10 - 3 3.23\times 10 - 3 3.12\times 10 - 3

32 1.72\times 10 - 3 1.44 1.36\times 10 - 3 1.25 1.40\times 10 - 3 1.1564 5.54\times 10 - 4 1.64 4.56\times 10 - 4 1.58 5.57\times 10 - 4 1.34128 1.55\times 10 - 4 1.84 1.29\times 10 - 4 1.82 1.73\times 10 - 4 1.68256 4.05\times 10 - 5 1.93 3.42\times 10 - 5 1.92 4.73\times 10 - 5 1.87512 1.03\times 10 - 5 1.97 8.76\times 10 - 6 1.96 1.22\times 10 - 5 1.94

As discussed in section 4.1, the fast algorithm in [19] can be applied to EFMto accelerate the computation. In (4.6), the integration on [0, \pi ) can be reduced to[0, \pi /2) and M is equal to the number of samples within [0, \pi /2). Table 2 presentsthe \ell 1 error for multiple values of M ; notice that M = 2 is good enough in practice,while in [19] the authors suggest M \geq 4.

Table 2The \ell 1 error of EFM with fast algorithm in [19] for multiple choices of N and M .

N M = 2 M = 3 M = 3216 4.6852\times 10 - 3 4.6826\times 10 - 3 4.6830\times 10 - 3

32 1.7241\times 10 - 3 1.7244\times 10 - 3 1.7245\times 10 - 3




In order to demonstrate that the proposed method satisfies the H-theorem nu-merically, we define a time-dependent discrete entropy function

(4.14) \eta (t) =

\biggl( 2T

N

\biggr) d \sum q\in X

Fq(t) lnFq(t).

The evolution of the entropy, plotted in Figure 3, shows that as the number of discretepoints N increases, the discrete entropy converges to the one of the exact solution.

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0 0.2 0.4 0.6 0.8 1

-4.3

-4.2

-4.1

-4

-3.9

Reference

N=32

N=64

N=128

N=256

Fig. 3. The evolution of the entropy of EFM for multiple N values.

-3 -2 -1 0 1 2 30

0.02

0.04

0.06

0.08

0.1

0.12

Exact solution

EFM

PPSM

Fig. 4. Comparison between PPSM and EFM at time t = 1 with N = 64 for the BKW solution.

As a comparison with PPSM, Figure 4 presents the numerical solutions in the v1direction of PPSM and EFM at t = 1 with N = 32. The smoothing filter used forEFM results in much less dissipation, thus leading to much better agreement withthe exact solution. Finally, Figure 5 shows that as N increases, the solution of EFMconverges rapidly to the exact solution.

Example 2 (bi-Gaussian initial value). Another frequent example is a problemwith the bi-Gaussian initial value

(4.15) f(t = 0, v) =1

4\pi

\biggl( exp

\biggl( - | v - u1| 2

2

\biggr) + exp

\biggl( - | v - u2| 2

2

\biggr) \biggr) ,

where u1 = ( - 2, 0)\sansT and u2 = (2, 0)\sansT . This is solved for the Maxwell molecules (2D invelocity) with radius R = 6. Figure 6 shows the numerical results of PPSM and EFM.The reference solution is calculated by the Fourier spectral method with N = 400 andR = 8. It is clear that the EFM solution is much closer to the reference solution.Figure 7 demonstrates that as N increases, EFM converges rapidly to the referencesolution.

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-3 -2 -1 0 1 2 30

0.02

0.04

0.06

0.08

0.1

Exact solution

N=32

N=64

N=128

-5 0 50

1

2

3

4

510

-4

Exact solution

N=32

N=64

N=128

Fig. 5. Numerical solution of EFM for multiple N values with the BKW solution at time t = 1on different scales.

-6 -4 -2 0 2 4 6

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Reference

EFM

PPSM

Fig. 6. Comparison between PPSM and EFM at t = 1 with N = 64 for the bi-Gaussian initialvalue.

Example 3 (discontinuous initial value). The initial condition given by

(4.16) f(t = 0, v) =

\left\{ \rho 1

2\pi T1 exp\Bigl( - | v| 2

2T1

\Bigr) if v1 > 0,

\rho 2

2\pi T2 exp\Bigl( - | v| 2

2T2

\Bigr) if v1 < 0

in this example is discontinuous. Here \rho 1 = 65 and the values of \rho 2, T1, and T2 are

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-3 -2 -1 0 1 2 30.03

0.04

0.05

0.06

0.07

Reference

N=32

N=64

N=128

-5 0 50

1

2

3

4

510

-5

Reference

N=32

N=64

N=128

Fig. 7. Numerical solution of EFM for multiple N values with the bi-Gaussian initial value attime t = 1 on different scales.

-6 -4 -2 0 2 4 6

0

0.05

0.1

0.15

0.2

0.25

0.3

(a) Profile of F (t = 0.5, v1, v2 = 0)

0

-4

0.05

0.1

-2

0.15

40

0.2

2

0.25

02-2

4 -4

(b) Profile of F (t = 0.5, v)

Fig. 8. Profile of F (t, v) with the discontinuous initial value (4.16) at time t = 0.5.

uniquely determined by the following conditions:\int \BbbR 2

f(0, v) dv =

\int \BbbR 2

f(0, v)| v| 2/2 dv = 1,

\int \BbbR 2

f(0, v)v dv = 0.

The profile of the reference solution is presented in Figure 8, which is computed

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-5 0 50

0.05

0.1

0.15

0.2

0.25

0.3

Reference

FGM

EFM

(a) N = 64

-1 0 10.05

0.1

0.15

0.2

0.25

0.3

Reference

FGM

EFM

(b) N = 64

-5 0 5-1

-0.5

0

0.5

1

1.5

210

-3

Reference

FGM

EFM

(c) N = 64

-5 0 50

0.05

0.1

0.15

0.2

0.25

0.3

Reference

FGM

EFM

(d) N = 128

-1 0 10.05

0.1

0.15

0.2

0.25

0.3

Reference

FGM

EFM

(e) N = 128

-5 0 5-5

0

5

1010

-4

Reference

FGM

EFM

(f) N = 128

-5 0 50

0.05

0.1

0.15

0.2

0.25

0.3

Reference

FGM

EFM

(g) N = 256

-1 0 10.05

0.1

0.15

0.2

0.25

0.3

Reference

FGM

EFM

(h) N = 256

-5 0 5-2

-1

0

1

2

3

4

10-4

Reference

FGM

EFM

(i) N = 256

Fig. 9. Numerical solution of EFM and FGM for multiple N values with the discontinuousinitial value (4.16) at time t = 0.5 on different scales.

by EFM with N = 2048. Due to the discontinuity in the initial value, the spectralaccuracy of FGM is lost. In addition, the Gibbs phenomenon leads to oscillations inthe initial value of FGM. In Figure 9, the plots around the discontinuity demonstratethat EFM has much better agreement as compared to FGM. The oscillations in FGMsolutions exhibit large errors, and the amplitude of the oscillation decreases slowly asN increases. On the contrary, there is no oscillation for EFM around the discontinuity,and the solution is always nonnegative.

Example 4 (3D BKW solution). The solution of this example is the exact 3DBKW solution, given by

(4.17) f(t, v) =1

(2\pi S)3/2exp

\biggl( - | v| 2

2S

\biggr) \biggl( 5S - 3

2S+

1 - S

2S2| v| 2

\biggr) ,

where S = 1 - 2 exp( - t/6)/5. Similarly to the 2D case, we first check the accuracy ofEFM. At time t = 0.01, the \ell 1, \ell 2, and \ell \infty errors and the convergence rates are listedin Table 3. Similarly to the 2D case, the convergence rate is of the second order andthe errors are rather small.

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Table 3The \ell 1, \ell 2, and \ell \infty errors and convergence rates for the BKW solution at time t = 0.01 with

R = 6.

N \ell 1 error Rate \ell 2 error Rate \ell \infty error Rate16 4.08\times 10 - 3 3.08\times 10 - 3 3.56\times 10 - 3

32 1.42\times 10 - 3 1.52 1.12\times 10 - 3 1.47 1.26\times 10 - 3 1.5064 4.07\times 10 - 4 1.80 3.29\times 10 - 4 1.76 3.72\times 10 - 4 1.76128 1.08\times 10 - 4 1.91 8.85\times 10 - 5 1.90 1.00\times 10 - 4 1.89

-2 -1 0 1 20.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

ExactSolution space

EFM N=32

EFM N=64

PPSM N=32

PPSM N=64

-6 -4 -2 0 2 4 60

1

2

3

4

510

-4

ExactSolution space

EFM N=32

EFM N=64

PPSM N=32

PPSM N=64

Fig. 10. Numerical solution of EFM and PPSM for multiple N values with the BKW solutionat time t = 1 on different scales.

As a comparison with PPSM, Figure 10 presents the numerical solutions on thev1 direction of PPSM and EFM at t = 1 with N = 32. The plots clearly show that thesmoothing filter used in EFM results in much less dissipation, thus leading to betteragreement with the exact solution.

5. Discussion. The EFM proposed in this paper is a tradeoff between accuracyand preservation of physical properties. The resulting scheme can be viewed as both adiscrete velocity method and a Fourier method. In terms of the convergence rate, it isbetter than DVM but slower than FGM. In terms of physical properties, it guaranteespositivity, mass conservation, and a discrete H-theorem, while the momentum andenergy conservation are lost. Regarding the computational cost, fast algorithms in[19, 11] remain valid for EFM. As for future work, we plan to study how to mitigatemomentum and energy loss, where higher order accuracy is needed for long timesimulation. The numerical implementation of the spatially inhomogeneous setting isalso in progress.

Acknowledgment. The authors thank Jingwei Hu for discussion on the filterand the fast algorithm on the Boltzmann collision term.

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biggl( vol. 40, no. 5, pp. a2858--a2882lexing/entropy.pdf · 2018. 10. 17. · entropic fourier...

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